Finite Group Databasebeta

← #30753251

Subgroups — #30753251

Order 17152.

450 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22565421C211yesmaximal, characteristic
2#22560205C211yesmaximal, characteristic
3#22558893C212yesmaximal
4#22558111C212yesmaximal
5C67×#2983C211yesmaximal, characteristic
6C2×#14524043C2211yescharacteristic
7#16863420C2212yes—
8C2×#14523962—21no—
9#16863335C414yes—
10#16863303—21no—
11#16863302—22no—
12#16863261—21no—
13C4×C4⋊Dic67C2211yescharacteristic
14#16863165—21no—
15#16863143—24no—
16C2×C4×Q8×C67C2211yescharacteristic
17#446×C67—24no—
18#432×C67—22no—
19#431×C67C2212yes—
20C22×Dic134—21no—
21C2×#11761612—22no—
22#14524043—22no—
23#14524043—22no—
24C2×C4⋊Dic67—21no—
25C2×C4⋊Dic67—21no—
26C2×C4⋊Dic67C2×C412yes—
27C2×C4×Dic67—21no—
28C4⋊(C67⋊C8)—22no—
29C4⋊(C67⋊C8)—22no—
30#14523962—42no—
31C67⋊#95—24no—
32C67⋊#95—24no—
33C67⋊#95C2×C412yes—
34C22×Q8×C67D411yescharacteristic
35C4×Q8×C67—22no—
36C4×Q8×C67—22no—
37C2×C4⋊C4×C67C2×C412yes—
38C2×C4⋊C4×C67D411yescharacteristic
39C2×C42×C67D412yes—
40C2×C42×C67C2311yescharacteristic
41C4⋊C8×C67—24no—
42C4⋊C8×C67D414yes—
43C22⋊C8×C67—22no—
44#97×C67—24no—
45C4×C8×C67—22no—
46C2×Dic134—44no—
47C2×Dic134—41no—
48C2×Dic134—41no—
49C22×Dic67—22no—
50C22×Dic67—22no—
51#11761612—42no—
52#11761612—42no—
53C2×C67⋊C8—42no—
54C4⋊Dic67—24no—
55C4⋊Dic67—42no—
56C4⋊Dic67—42no—
57C4×Dic67—41no—
58C4×Dic67—41no—
59C2×Q8×C67—22no—
60C2×Q8×C67—21no—
61C2×Q8×C67—21no—
62C2×Q8×C67—22no—
63C22×C4×C67C22⋊C412yes—
64C22×C4×C67C22⋊C412yes—
65C22×C4×C67C22×C411yescharacteristic
66C22×C4×C67C2×D411yescharacteristic
67C22×C4×C67C2×D411yescharacteristic
68C2×C8×C67—24no—
69C4⋊C4×C67—24no—
70C4⋊C4×C67—22no—
71C42×C67—22no—
72C42×C67—22no—
73C42×C67C2×D412yes—
74C42×C67#4112yes—
75C23×C67C2×C22⋊C411yescharacteristic
76Q8×C67—42no—
77Q8×C67—42no—
78C2×C4×C67—22no—
79C2×C4×C67—22no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67—41no—
84C2×C4×C67—42no—
85C2×C4×C67C4×D412yes—
86C2×C4×C67#12011yescharacteristic
87C2×C4×C67#12112yes—
88C2×C4×C67#12311yescharacteristic
89C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
90C2×C4×C67#13712yes—
91C2×C4×C67#13711yescharacteristic
92C2×Dic67—42no—
93C2×Dic67—42no—
94C2×Dic67—44no—
95C2×Dic67—42no—
96C2×Dic67—42no—
97Dic134—82no—
98Dic134—82no—
99C8×C67—44no—
100C67⋊C8—82no—
101C22×C67—21no—
102C22×C67—21no—
103C22×C67#38611yescharacteristic
104C22×C67#41711yescharacteristic
105C22×C67#41911yescharacteristic
106C4×C67—21no—
107C4×C67—22no—
108C4×C67—42no—
109C4×C67—42no—
110C4×C67#45412yes—
111C4×C67#46812yes—
112C4×C67#47012yes—
113Dic67—82no—
114Dic67—82no—
115#9404—671nomaximal, Sylow
116C2×C67—21no—
117C2×C67—21no—
118C2×C67#188811yescharacteristic
119C2×C67#189411yescharacteristic
120C2×C67#191511yescharacteristic
121#2983D6711yescharacteristic
122#2568—671no—
123#1801—671no—
124#1549—672no—
125#1303—672no—
126C67#940411yescharacteristic, complemented, Sylow
127C2×C4⋊Q8—671no—
128C2×C4×Q8C2×D6711yescharacteristic
129#446—24no—
130#432—22no—
131#431C2×D6712yes—
132#423—672no—
133C2×#101—1341no—
134#400—674no—
135#391—1341no—
136#391—1342no—
137#382—1341no—
138C4×C4⋊C4—671no—
139#343—1341no—
140#332—1344no—
141C22×Q8—1341no—
142C22×Q8D26811yescharacteristic
143C2×M16—1342no—
144C4⋊Q8—1342no—
145C4⋊Q8—1342no—
146C4×Q8—22no—
147C4×Q8—22no—
148C2×C4⋊C4—672no—
149C2×C4⋊C4—1341no—
150C2×C4⋊C4—1341no—
151C2×C4⋊C4C4×D6712yes—
152C2×C4⋊C4D26811yescharacteristic
153C2×C42—1341no—
154C2×C42C22⋊D6712yes—
155C2×C42C22×D6711yescharacteristic
156C4⋊C8—24no—
157C4⋊C8—1342no—
158C4⋊C8—1342no—
159C4⋊C8C22⋊D6714yes—
160#101—2682no—
161C22⋊C8—22no—
162#97—24no—
163C4×C8—22no—
164#95—672no—
165#95—1344no—
166#95—1344no—
167C2×Q8—22no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—22no—
171C2×Q8—2684no—
172C2×Q8—2681no—
173C2×Q8—2681no—
174C22×C4—1342no—
175C22×C4—1342no—
176C22×C4(C2×D67)⋊C412yes—
177C22×C4(C2×D67)⋊C412yes—
178C22×C4C2×C4×D6711yescharacteristic
179C22×C4C2×C22⋊D6711yescharacteristic
180C22×C4C2×D26811yescharacteristic
181M16—2682no—
182M16—2682no—
183C2×C8—24no—
184C2×C8—2682no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—2682no—
188C4⋊C4—2682no—
189C4⋊C4—1344no—
190C42—22no—
191C42—22no—
192C42—2681no—
193C42—2681no—
194C42C2×C22⋊D6712yes—
195C42#1176162812yes—
196C23C2×(C2×D67)⋊C411yescharacteristic
197Q8—42no—
198Q8—42no—
199Q8—5362no—
200Q8—5362no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—2682no—
209C2×C4—2684no—
210C2×C4—2682no—
211C2×C4—2682no—
212C2×C4—2682no—
213C2×C4C4×C22⋊D6712yes—
214C2×C4#1452401211yescharacteristic
215C2×C4#1452401712yes—
216C2×C4#1452402611yescharacteristic
217C2×C4#1452407111yescharacteristic
218C2×C4#1452407612yes—
219C2×C4#1452407611yescharacteristic
220C8—44no—
221C8—5362no—
222C22—21no—
223C22—21no—
224C22#1686327511yescharacteristic
225C22#1686339811yescharacteristic
226C22#1686340611yescharacteristic
227C4—21no—
228C4—22no—
229C4—42no—
230C4—42no—
231C4—5362no—
232C4—5362no—
233C4#1686356912yes—
234C4#1686364212yes—
235C4#1686365412yes—
236C2—21no—
237C2—21no—
238C2#2256061211yescharacteristic
239C2#2256064211yescharacteristic
240C2#2256076111yescharacteristic
subgroupG/H
centerC22#16863275
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2983C2
socleC22×C67#386
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C42×C67C23
hypercenter#2983D67
hypocenterC67#9404
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
126. C67#9404semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225654212
#225602052
#225588932
#225581112
C67×#29832
#940467